1.7 Universal Set

NCERT Class 11 Mathematics Textbook for blind and visually impaired students made screen readable by Professor T K Bansal.

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context.


For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth.


This basic set is called the “Universal Set”. The universal set is usually denoted by the letter capital U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of all rational numbers or, for that matter, the set R of all real numbers.

For another example, in human population studies, the universal set consists of all the people in the world.

EXERCISE 1.3

Question 1.


Make correct statements by filling in the symbols ⊂ or ⊄ in the lank spaces :


(i) { 2, 3, 4 } ... { 1, 2, 3, 4, 5 }


(ii) { a, b, c } ... { b, c, d }


(iii) {x : x is a student of Class 11 of your school} ...{x : x is a student of your school}


(iv) {x : x is a circle in a plane} ...{x : x is a circle in the same plane with radius 1 unit}


(v) {x : x is a triangle in a plane} ... {x : x is a rectangle in the same plane}


(vi) {x : x is an equilateral triangle in a plane} ... {x : x is a triangle in the same plane}


(vii) {x : x is an even natural number} ... {x : x is an integer}

Answer 1.


(i) ⊂


(ii) ⊄


(iii) ⊂


(iv) ⊄


(v) ⊄


(vi) ⊂


(vii) ⊂

Question 2.


Examine whether the following statements are true or false:


(i) { a, b } ⊄ { b, c, a }


(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}


(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }


(iv) { a } ⊂ { a, b, c }


(v) { a }∈ { a, b, c }


(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}

Answer 2.


(i) False


(ii) True


(iii) False


(iv) True


(v) False


(vi) True

Question 3.


Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?


(i) {3, 4} ⊂ A


(ii) {3, 4} ∈ A


(iii) {{3, 4}} ⊂ A


(iv) 1 ∈ A


(v) 1 ⊂ A


(vi) {1, 2, 5} ⊂ A


(vii) {1, 2, 5} ∈ A


(viii) {1, 2, 3} ⊂ A


(ix) φ ∈ A


(x) φ ⊂ A


(xi) {φ} ⊂ A

Answer 3.


(i) as {3,4}∈A,


(v) as 1 ∈ A,


(vii) as {1, 2, 5}⊂A,


(viii) as 3∉A,


(ix) as φ ⊂ A,


(xi) as φ ⊂ A

Question 4.


Write down all the subsets of the following sets


(i) {a}


(ii) {a, b}


(iii) {1, 2, 3}


(iv) φ

Answer 4.


(i) φ , { a }


(ii) φ , { a }, { b }, { a, b }


(iii) φ, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 }


(iv) φ

Question 5.


Write the following as intervals :


(i) {x : x ∈ R, − 4 < x ≤ 6}


(ii) {x : x ∈ R, − 12 < x < −10}


(iii) {x : x ∈ R, 0 ≤ x < 7}


(iv) {x : x ∈ R, 3 ≤ x ≤ 4}

Answer 5.


(i) (− 4, 6]


(ii) ( −12, −10)


(iii) [ 0, 7 )


(iv) [ 3, 4 ]

Question 6.


Write the following intervals in set-builder form :


(i) ( −3, 0)


(ii) [6 , 12]


(iii) (6, 12]


(iv) [ −23, 5)

Answer 6.


(i) { x : x ∈ R, − 3 < x < 0 }


(ii) { x : x ∈ R, 6 ≤ x ≤ 12 }


(iii) { x : x ∈ R, 6 < x ≤ 12 }


(iv) { x R : − 23 ≤ x < 5 }

Question 7.


What universal set(s) would you propose for each of the following :


(i) The set of right triangles.


(ii) The set of isosceles triangles.

Question 8.


Given the sets


A = {1, 3, 5},


B = {2, 4, 6} and


C = {0, 2, 4, 6, 8},


which of the following may be considered as universal set (s) for all the three sets A, B and C


(i) {0, 1, 2, 3, 4, 5, 6}


(ii) φ


(iii) {0,1,2,3,4,5,6,7,8,9,10}


(iv) {1,2,3,4,5,6,7,8}

Answer 8.


(iii)